WikiProject Mathematics (Rated C-class, High-importance)
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Field: Algebra

## ??

Perhaps it would also be helpful to add the case of one variable? I say this because most of the general public will deal only with this case in high school and the first year of college calculus. Like so:)

$F(x) = ax^2 + bx + c$

However I don't feel quite comfortable enough mathematically to actually modify the article, so if someone smart would like to implement this, I'd enjoy that :)

Goodralph 10:15, 3 Mar 2004 (UTC)

Charles Matthews 11:07, 3 Mar 2004 (UTC)

Added comment to that effect to article. - dcljr 06:51, 24 Feb 2005 (UTC)

Indeed quadradic forms must be homogeneous of degree 2 i.e. the sum of the exponents in each term must be 2.

Indeed quadradic forms must be homogeneous of degree 2 i.e. the sum of the exponents in each term must be 2.

We need a either a section or an article on the properties of quadratic forms used in statistics. There are about a half dozen important theorems about these. For example, if $g$ is a vector of constants, $\epsilon$ is a random vector whose entries are independent with variance $\sigma^2$, and $y=g+\epsilon$, then $\operatorname{E}\left[y'Ay\right]=\sigma^2\operatorname{tr}\left[A\right]+g'Ag$. I think there is enough to warrant a separate article on this; are there any objections? Btyner 18:41, 27 November 2005 (UTC)

## error in isotropic defn?

The definition of isotropic and anisotropic in this article appears to be reversed, at least to me. It defines an "isotropic space" as one whose form has a non-trivial kernel. Surely such a space should be anisotropic, instead? That defn has been there a longgg time. linas 13:11, 19 July 2006 (UTC)

The definition in the article seems to agree with J. P. Serre in "A Course in Arithmetic" (interesting title!), so Isuspect it might be right. Madmath789 14:57, 19 July 2006 (UTC)

The definition is correct. Isotropic is the standard name for spaces with non zero elements $v$ such that $Q(v)=0$. This is according to the refrence listed on the bottom of the article.

Perhaps this is a development more recent the Serre's book, but the definition I've seen for isotropic has to do with the the associated bilinear form:
• v is isotropic if b(v,v)=0
and the other definition is
• v is singular if Q(v)=0
Note in odd characteristic these definitions are the same because the quadratic form is completely determined by its' bilinear form. However, in even characteristic there are isotropic vectors that are not singular. These references may also handle the

char != 2 case first and then revamp it later. IMHO there is something very important missing here. —Preceding unsigned comment added by 98.30.181.0 (talk) 15:49, 16 April 2010 (UTC)

## Start out simple

Very few readers will be interested in topology and number theory. Therefore these sections should be moved to a section near the end or possibly transferred to a separate article.

## Diagram

There should be a diagram of the quadratic form of a matrix. —Ben FrantzDale 22:30, 19 January 2007 (UTC)

## Error in definiteness definition

To me there appears to be an error in the leading principal minor definition of positive and negative definiteness:

According to Robert A. Adams Calculus - A complete course 6th Edition, Section 10.6 Theorem 8 (p. 579): (Where A symmetric n×n matrix, Di denotes the principal leading minor of size i×i)

a) If Di>0 for 1≤i≤n, then A is positive definite
b) If Di>0 for even numbers i in {1,2,…,n}, and Di<0 for odd numbers i in {1,2,…,n}, then A is negative definite …

b) above seems to conflict with the first statement concerning principal leading minors in the article (which says that A is negative definite if Di<0 for each i)

Hopefully someone who knows these things more clearly than me can edit the article (I was actually just looking for a proof of (the unproven) theorem 8 in Adams' book) Tinwelinto 20:30, 7 March 2007 (UTC)

You're right. I just deleted the bit about negative definite matrices. It is not so hard to derive, if necessary, and seems not that important here. I also removed the statement "the real symmetric matrix is positive semidefinite if and only if it has all non-negative leading principal minors" in view of the counterexample
$\begin{bmatrix} 0 & 0 \\ 0 & -1 \end{bmatrix}.$
All two leading principal minors are zero, but the matrix is not positive semidefinite. -- Jitse Niesen (talk) 08:19, 12 March 2007 (UTC)

Why does the field that the vector space is defined over have to be real for positive definiteness to hold? Surely the quadratic form on C^n given by the n x n identity matrix is positive definite. —Preceding unsigned comment added by 82.14.70.59 (talk) 23:03, 12 January 2008 (UTC)

## Only for the finite-dimensional case

"Bilinear forms are the full tensor product $V^* \otimes V^*$,..." well, this is only if V is finite-dimensional, right? Otherwise we have to write $(V \otimes V)^*$ instead, correct? Commentor (talk) 04:50, 3 March 2008 (UTC)

## being more careful (or picky or pedantic) about the definition

That is, i think it is confusing to write expressions like

$F(x,y,z) = ax^2 + by^2 + cz^2 + dxy + exz + fyz$

and also write

$B(u,v) = Q(u+v) - Q(u) - Q(v)$

with the idea that Q and F are the same sorts of things (quadratic forms) while B is something different (a bilinear form).

If it can be done in a non-clumsy way, i think it would be better to make it very clear just what we think a quadratic form is (e.g., a certain kind of function of one vector-valued variable, if we are thinking of these things as functions rather than merely formal polynomials), and use notation which suggests this.

My complaint sounds petty because after all if you spend some time thinking about it the meaning of the article is clear.

But the article is intended for a very broad audience and is not for publication in a journal. So i think it would be good to give up some of the minimality in exchange for a little more precision.

It is a good article, and my criticism, even if it is correct, is small. —Preceding unsigned comment added by 99.23.189.19 (talk) 06:38, 10 April 2009 (UTC)

## Identities

Moved over from the main article: section "Identities". I cannot immediately see if we have an article where this section will fit, but it certainly doesn't fit in here. Arcfrk (talk) 23:42, 22 September 2009 (UTC)

### Identities

Starting with one solution {y1, y2, y3} of the diagonal form,

$ay_1^2+by_2^2+cy_3^2 = 0$

an infinite more can be given using the identity,

$ax_1^2+bx_2^2+cx_3^2 = (ay_1^2+by_2^2+cy_3^2)(az_1^2+bz_2^2+cz_3^2)^2$

where the xi are,

$[x_1, x_2, x_3] = [uy_1-vz_1, uy_2-vz_2, uy_3-vz_3] \,$, and,
$[u, v] = [az_1^2+bz_2^2+cz_3^2, 2(ay_1z_1+by_2z_2+cy_3z_3)]$

for arbitrary {z1, z2, z3}. Similarly, for quaternary diagonal forms,

$ax_1^2+bx_2^2+cx_3^2+dx_4^2 = (ay_1^2+by_2^2+cy_3^2+dy_4^2)(az_1^2+bz_2^2+cz_3^2+dz_4^2)^2$

where the xi are,

$[x_1, x_2, x_3, x_4] = [uy_1-vz_1, uy_2-vz_2, uy_3-vz_3, uy_4-vz_4] \,$, and,
$[u, v] = [az_1^2+bz_2^2+cz_3^2+dz_4^2, 2(ay_1z_1+by_2z_2+cy_3z_3+dy_4z_4)]$

for arbitrary {z1, z2, z3, z4}, and so on for any number of addends xi.[1]

## Quadratic forms defined by a non-symmetric matrix

### Why my contribution was deleted without any attempt of consensus?

You are not encouraging new editors.

Arcfrk: if you find it "unhelpful/confusing", why didn't you ask first for a better solution. We can clarify or expand the remark. It is a sufficiently relevant fact.

What if a revert your change arbitrarily as you did? Wikipedia can't work that way.

Francisco Albani (talk) 01:32, 29 November 2009 (UTC)

I would certainly prefer to see this discussed at symmetric bilinear form. It is clearer to understand the relationship between general and symmetric bilinear forms as one step, with the remark that the semi-sum symmetrization depends on being able to divide by 2 in the field; and then the relationship between symmetric bilinear forms and quadratic forms as another step. If you can divide by 2 freely there is no problem; but that is not always the case in this subject. Charles Matthews (talk) 09:25, 29 November 2009 (UTC)

## Alternative definition when 2 is not invertible

My revision http://en.wikipedia.org/w/index.php?title=Quadratic_form&oldid=354625703#Quadratic_forms was undid. Here's the definition in the case of a vector space over field F with reference. It seems to me that they should correspond to the definition in the module case, these things almost always do. If not please provide a source, I'd like to see it :)

Q:V -> F is a quadratic form if

• Q(av)=a^2Q(v)
• b(u,v)=Q(u+v)-Q(u)-Q(v) is bilinear

(Taylor 1992, page 54)

When F has characteristic not equal to 2 then Q(v)=b(v,v)/2 (Taylor 1992, page 55)

• Taylor, Donald E. (1992), The Geometry of the Classical Groups 9, Berlin: Heldermann Verlag, ISBN 3-88538-009-9, MR 1189139 Text " Sigma Series in Pure Mathematics " ignored (help)

PS: Given enough time to visit the library I can provide further reference if you have trouble acquiring Donald Taylor's book. —Preceding unsigned comment added by 98.30.181.0 (talkcontribs)

There is nothing wrong with the definition itself. I've reverted it mainly because it leads to a different notion of the associated bilinear form, as compared with the next section, so it was just a quick consistency fix. I've now looked at a couple of standard sources for the algebraic theory of quadratic forms (Scharlau and Pfister), and they treat the case of characteristic ≠2 first, and characteristic 2 separately and somewhat later, and the associated bilinear form is defined differently! Earlier I commented that it was unwise to go for the ultimate generality and discuss quadratic forms over rings; but now I see that even over the fields, separate definitions are necessary for char 2.
Some time ago I started to streamline the presentation in this article and (re)wrote the first few sections, but ran out of steam and haven't tidied up the rest. The "definition" section and the next section need to be be merged and cleaned up. Right now, they are inaccurate, confusing, and redundant. For this reason, any changes made now will likely be transitory anyway. Arcfrk (talk) 06:41, 10 April 2010 (UTC)

Okay seems perfectly reasonable to me. I will certainly have some time to help with the streamlining / improving of the article in the next couple months. Admittedly my focus on quadratic forms has been mainly in the characteristic 2 case over fields. It is a little discouraging, but not surprising, that the definitions are somewhat inconsistent. I'm sure if we pool our efforts we can bring this article into better shape without too much trouble. I suppose it probably would be wise if I created a wikipedia account :) —Preceding unsigned comment added by 98.30.181.0 (talk) 16:06, 10 April 2010 (UTC)

## non-singular or non-degenerate quadratic form

I don't think non-singular is defined correctly for the quadratic form. Orthogonal_group defines the orthogonal group to be linear transformations that preserve some non-singular quadratic form. So I assume this is what's called a non-degenerate quadratic form in the literature I've read. Here's how I've seen it before:

Well, since q(x) = b(x,x), the restriction of the quadratic form q to the kernel of b is identically zero. So the only way for ker b to be anisotropic is if it only contains zero vector, i.e. ker b = 0. Do you have a reference? Does it have anything to do with characteristic 2 or modules over a ring? In both cases, the definitions of the associate bilinear form and of "non-degenerate" need to be modified. Arcfrk (talk) 03:51, 13 April 2010 (UTC)

Unfortunately q(x)=b(x,x) only when char !=2. In characteristic 2 there is a difference. In fact there are bilinear forms in characteristic 2 that are not the associated bilinear form to any quadratic form :) —Preceding unsigned comment added by 98.30.181.0 (talk) 15:51, 16 April 2010 (UTC)

Sorry thought I was signed in above. Is my comment —Preceding unsigned comment added by Somethingcompletelydifferent (talkcontribs) 15:55, 16 April 2010 (UTC)

Well, since you agree that it's a characteristic 2 issue, why don't you write a section dedicated to char 2 case, as you previously indicated you wanted to? It is certainly needed for the completeness of coverage! Think it through and give all the necessary definitions and theorems. In the meantime, please, do not insert definitions/statements inconsistent with the rest of the text. All of the present section is dealing with characteristic NOT 2, as prominently displayed just a short while before. In particular, the term "associate bilinear form" has a precise meaning, q(x)=b(x,x), and your way of phrasing the definition of nonsingular simply does not work in this context. Arcfrk (talk) 06:02, 24 April 2010 (UTC)

## "intentional" space

Arcrfk, you insist on keeping a space at the beginning of a line to set the following in typewriter text: "Let us assume that the characteristic of K is different from 2." If you feel strongly that it should be set in its own text box, fine, but there is no reason to set it in typewriter text. It is completely nonstandard within Wikipedia. I don't see why it can't just be its own line without anything special to set it off, but setting it typewriter text is just wrong. —Ben FrantzDale (talk) 14:15, 22 October 2010 (UTC)

## geometric meaning

Can someone related the quadratic form to geometric meaning. when it is positive definite, it is ellipsoid? what condition corresponds to hyperboloid (elliptic or parabolic) or paraboloid???

Suppose the equation is $x^TAx=1$, then it seems that

if the matrix can be turned into a diagonal matrix, then it is an ellipsoid or a hyperboloid. If all the eigen values are non negative, then it is an ellipsoid, if some eigen values are negative, then it is a hyperboloid. If the matrix can not be turned into a diagonal one, then it is a paraboloid (either elliptic or hyperbolic). If all the eigen values are non negative, then it is elliptic, if some eigen values are negative, then it is hyperbolic).

Jackzhp (talk) 14:55, 13 February 2011 (UTC)

## "Jacobi's Theorem

This article mentions "Jacobi's Theorem" under "Real Quadratic Forms" but doesn't link that to anywhere and when I search for jacobi's theorem in wikipedia, I don't see a disambiguation link for this use of the term.

Bill Smith (talk) 19:17, 20 January 2013 (UTC)

It refers to this sentence a few lines earlier:
A fundamental theorem due to Jacobi asserts that q can be brought to a diagonal form
Deltahedron (talk) 19:49, 20 January 2013 (UTC)

## Citations

I was not sure this article really must have additional citations. I think it was basic stuff from the authors point of view and not especially dubious at any points. But I admit I am new to this topic so that can be factored into my opinion. Further I would like to mention that over dependence on citations can be meaningless. For example, you can find a million citations that support significant man caused global warming but that may actually be negative evidence. Pardon the political reference and substitute any example where citations may not be accurate. However if there is something truly dubious here let's get to that issue. § — Preceding unsigned comment added by Berrtus (talkcontribs) 08:23, 15 April 2013 (UTC)

While lending credibility to the correctness of information is an important function of citations, that's actually secondary to the more fundamental use: showing the reader where the information appears. This is just an encyclopedia, so its role is to act as a reference to reputable sources (sometimes despite the sources' correctness or incorrectness.) In any case, having a citation to a good reference is invaluable to a reader who wants to learn more. Having the "needs more citations" flag doesn't always have to be interpreted as a black eye! I think in this case, the contents of the article is OK: however, it looks like someone wished they had some more detailed pointers to where they could continue reading :) Rschwieb (talk) 13:47, 15 April 2013 (UTC)
I have added various items as a Further Reading section. Deltahedron (talk) 16:51, 15 April 2013 (UTC)